Blind frequency-offset estimation for spatially correlated signal

ABSTRACT

An iterative, blind, frequency-offset estimation process that does not require any training signal or demodulated information symbols is disclosed. Receivers embodying the disclosed processes can produce periodic frequency-offset estimates, without running computationally intensive equalization or demodulation algorithms, by exploiting the temporal correlation of the received signal in the time domain, as well as the received signal&#39;s correlation across in-phase and quadrature dimensions, in some embodiments, to find a frequency-offset estimate that best fits the received signal in a maximum-likelihood sense. In an exemplary method of estimating receiver frequency offset, a spatially stacked signal block is formed from multi-branch signal samples corresponding to each of two or more time-separated samples of the received signal. The spatially stacked signal block is used in computing a maximum-likelihood joint estimate of the receiver frequency offset and the spatial covariance of the spatially stacked signal block de-rotated by the receiver frequency offset.

This application is a continuation of pending application Ser. No.12/275,992, entitled “Blind Frequency-Offset Estimation for Temporallyand/or Spatially Correlated Signal”, filed Nov. 21, 2008.

TECHNICAL FIELD

The present invention relates to telecommunication systems, inparticular to methods and apparatus for estimating receiver frequencyoffset in wireless communication receivers.

BACKGROUND

In a cellular communication system, a key task of a mobile terminal isto synchronize its internal reference clock to the carrier frequency ofthe serving or other nearby base station. Frequency synchronization isneeded not only to enable proper reception of the radio signaltransmitted from the base station, but also to ensure that the frequencyof radio signals emitted from mobile terminal meets tight systemspecifications, so that substantial interference is not generated forother users.

To maintain a proper frequency reference for its oscillator, a mobileterminal typically performs periodic estimation of the frequency offset,i.e., the deviation of a local reference signal from the actualfrequency of the transmitted signal, based on a signal received from aserving or monitored base station. The resulting frequency-offsetestimates are used to adjust the reference frequency in the oscillatorto keep it from drifting away from the correct designated frequency, tocompensate digital signal processing performed on received signals, orboth. An efficient algorithm for accurately estimating the frequencyoffset from the received signal is thus essential to the normaloperation of a mobile terminal.

Many conventional frequency estimation algorithms require the receiverto have certain knowledge about the actual transmitted signal, whichmight be derived either through the use of a pre-determined trainingsignal or through the use of demodulated information symbols in adecision-directed manner. However, in many cases the training signal maybe too short for the receiver to derive an accurate frequency-offsetestimate, while decision-directed estimation can be computationallyexpensive and may require the use of specialized hardware acceleratorthat lacks the flexibility for future design enhancement.

Moreover, many conventional decision-feedback frequency-estimationalgorithms treat the baseband received signal as a complex-valuedsignal, thus treating in-phase and quadrature components of the receivedsignal as the real and imaginary parts, respectively, of acomplex-valued signal. These algorithms are generally derived assumingthat complex arithmetic operations are used throughout the receiver. Asa result, these algorithms are incompatible with receivers that treatthe in-phase and quadrature components of the received signal asseparate “spatial” dimensions, or branches, and apply more generaltwo-dimensional “spatial” operations on the received signal. Forexample, a GSM single-antenna interference cancellation (SAIC) receivertypically treats the in-phase and quadrature components of the receivedsignal as though they come from two different antenna elements. Thespatial operations involved in such receivers are essential to theinterference cancellation capability, but can also significantly distortthe phase information of the corresponding complex-valued signal.Consequently, specially designed algorithms are needed for estimatingfrequency offset in such receivers.

SUMMARY

Various embodiments of the present invention employ an iterative, blind,frequency-offset estimation process that does not require any trainingsignal or demodulated information symbols. As a result, receiversembodying the disclosed processes can produce periodic frequency-offsetestimates by monitoring data bursts that may be intended for otherusers, for example, without running computationally intensiveequalization or demodulation algorithms.

The disclosed receivers and processing techniques exploit the temporal(block) correlation of the received signal in the time domain, as wellas the received signal's correlation across in-phase and quadraturedimensions, in some embodiments, to find a frequency-offset estimatethat best fits the received signal in a maximum-likelihood sense. Thetemporal (block) correlation of the received signal may arise fromchannel dispersion, partial-response signaling of the underlying desiredsignal or a dominant interfering signal, or both, while correlationacross in-phase and quadrature components may arise from the presence ofa one-dimensional symbol constellation in the received signal, such aswith binary phase-shift keying (BPSK) modulation or Gaussianminimum-shift keying (GMSK) modulation.

The disclosed techniques may be applied to any of a variety ofmulti-branch receiver architectures designed for various cellularsystems, including GSM/EDGE systems, single-carrier frequency-divisionmultiple access (SC-FDMA) receivers in 3GPP's Long-Term Evolution (LTE)systems, or other orthogonal frequency-division multiplexing (OFDM)based systems.

Some embodiments of the present invention are based on a complex-valuedmodel of the baseband signal, and thus employ only complex arithmeticoperations. Other embodiments are based on a two-dimensional spatialmodel for the in-phase and quadrature components of the baseband signal,and use two-dimensional matrix operations on the in-phase and quadraturebranches. The former embodiments exploit temporal correlations of thereceived signal, while the latter also exploit the spatial correlationacross in-phase and quadrature domains. This latter approach can lead tosubstantial performance gain when the received signal's modulationconstellation is one-dimensional on the in-phase/quadrature plane.

In an exemplary method of estimating receiver frequency offset in acommunications receiver, a temporally stacked signal block is formedfrom multi-branch signal samples corresponding to each of two or moretime-separated samples of the received signal. The temporally stackedsignal block is used in computing a maximum-likelihood joint estimate ofthe receiver frequency offset and the spatial covariance of thetemporally stacked signal block de-rotated by the receiver frequencyoffset. In some embodiments, the temporally stacked signal block isgenerated by forming a vector by stacking complex-valued samples foreach branch of the multi-branch signal samples, while in otherembodiments the temporally stacked signal block may comprise a vectorformed by stacking real-valued in-phase and quadrature samples for eachbranch of the two or more multi-branch signals.

In some embodiments of the invention, computing a maximum-likelihoodjoint estimate of the receiver frequency offset and the spatialcovariance of the temporally stacked signal block de-rotated by thereceiver frequency offset is performed using an iterative process. Insome of these embodiments, a de-rotated received signal block iscomputed from the temporally stacked signal block in each iteration,using a current estimate of the receiver frequency offset. A signalcovariance matrix is estimated from the de-rotated received signalblock, for each iteration, and the current estimate of the receiverfrequency offset is updated as a function of the estimated signalcovariance matrix. This iterative process may continue for severaliterations until a pre-determined condition is met. For instance, insome embodiments, the process may simply be repeated for apre-determined maximum number of iterations. In other embodiments, alog-likelihood of the joint estimate of the receiver frequency offsetand the spatial covariance may be computed for each iteration, and theiterative process repeated until the incremental change in computedlog-likelihoods between successive iterations fall below apre-determined threshold.

Further embodiments of the present invention include a wireless receiver(which may be embodied in a wireless transceiver configured foroperation with one or more wireless standards) that includes one or moreprocessing circuits configured to carry out one or more of thefrequency-offset estimation techniques described herein. Of course,those skilled in the art will appreciate that the present invention isnot limited to the above features, advantages, contexts or examples, andwill recognize additional features and advantages upon reading thefollowing detailed description and upon viewing the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a wireless communication system including anexemplary wireless transceiver according to some embodiments of thepresent invention.

FIG. 2 illustrates an exemplary process flow for estimating receiverfrequency offset in a communications receiver.

FIG. 3 is another process flow diagram illustrating a method forestimating receiver frequency offset according to some embodiments ofthe present invention.

FIG. 4 illustrates details of another exemplary wireless transceiveraccording to some embodiments of the present invention.

DETAILED DESCRIPTION

The frequency-offset estimation techniques disclosed herein may beapplied to any of a variety of multi-branch receiver architecturesdesigned for various cellular systems, including GSM/EDGE systems,single-carrier frequency-division multiple access (SC-FDMA) receivers in3GPP's Long-Term Evolution (LTE) systems, or other orthogonalfrequency-division multiplexing (OFDM) based systems. Accordingly, FIG.1 illustrates a simplified block diagram of a wireless system 100,including a base station 110 and wireless transceiver 150. The basestation 110 may comprise, for example, a GSM/EDGE base station or an LTEevolved Node B, while wireless transceiver 150 contains wirelessreceiver circuits and wireless transmitter circuits configured forcompatibility with one or more wireless standards and one or morefrequency bands, including the wireless standard(s) and frequencyband(s) supported by base station 110.

As pictured, exemplary wireless transceiver 150 comprises two receiveantennas providing a multi-branch received signal to duplexing circuit152, which in turn provides the multi-branch received signal to receiverfront-end circuit 156. Those skilled in the art will appreciate thatduplexer circuit 152 may comprise one or more filters and/or switches,depending on the wireless standard, and serves to alternately connectthe antennas to the receiver (RX) front-end circuit 156 and thetransmitter (TX) analog section 154, or to isolate signals generated byTX analog section 154 from the RX front-end circuit 156, or both. Thoseskilled in the art will appreciate that TX analog section 154 and RXfront-end circuit 156 may comprise one or more power amplifiers,low-noise amplifiers, filters, mixers, modulators, analog-to-digitalconverters (ADCs), digital-to-analog converters (DACs), and the like,configured according to well-known techniques appropriate for thewireless standard(s) and frequency band(s) supported by wirelesstransceiver 150. Accordingly, further details of these circuits, whichdetails are not necessary for an understanding of the present invention,are not provided herein.

TX analog section 154 and RX front-end 156 are supplied with one or morereference signals from phase-locked loop (PLL) circuit 158, which inturn is driven by a reference clock oscillator 159, which may be atemperature-compensated crystal oscillator circuit, for example. Again,design details for various suitable phase-locked loop circuits andreference oscillator circuits are well known to those skilled in theart, and are not provided here.

Baseband signal processing circuit 160, which may comprise one orseveral microprocessors, microcontrollers, digital signal processors, orthe like, configured with appropriate firmware and/or software, suppliesa transmit baseband signal (which may comprise, for example, a series ofencoded digital samples) to TX analog section 154; TX analog section 154modulates a reference signal provided by PLL 158 with the transmitbaseband signal, up-converts the modulated signal to a radio-frequency(RF) carrier signal frequency, as necessary, and amplifies the RF signalfor transmission to base station 110. Baseband signal processing circuit160 also receives a receiver baseband signal from RX front-end circuit156, which may comprise, for example, a homodyne or heterodynedownconverter followed by one or more ADCs, and performs demodulationand decoding processes according to conventional techniques.

In the wireless transceiver 150 pictured in FIG. 1, complex-valued,multi-branch, received signal samples r_(c)[n] are provided by RXfront-end circuit 156 to the baseband signal process circuit 160, foruse, among other things, by frequency offset estimation circuit 165. Inthis embodiment, the temporal correlation of the multi-branch signalsamples can be exploited, as described in detail below, to estimate thefrequency error, or “frequency offset”, of the wireless receiver. Aswill also be described below in reference to FIG. 4, other embodimentsmay further exploit the spatial correlation between the in-phase andquadrature components of the received signal.

In any event, the reference signals used in wireless transceiver 150generally must be tightly controlled to match the correspondingfrequency sources at the base station 110. This tight control isnecessary to ensure that signals transmitted by the wireless transceiver150 remain within a pre-determined bandwidth, so as to avoid excessinterference to other wireless devices. This tight control is alsonecessary to ensure proper demodulation and decoding of the receivedsignals. Accordingly, baseband signal processing circuit 160 comprises afrequency-offset estimation circuit 165, which is configured to estimatethe frequency offset between the frequency references used by wirelesstransceiver 150 and those used by base station 110, according to one ormore of the techniques described herein. Those skilled in the art willappreciate that the term “frequency offset,” as used herein, is intendedto generally refer to the frequency error in reference signals used inthe wireless transceiver 150, as related to the carrier frequency (orfrequencies) transmitted by the base station 110 and received by thewireless transceiver 150. Those skilled in the art will appreciate thatfrequency offsets are commonly expressed as dimensionless relativequantities, e.g., 1×10⁻⁸ or 10 parts-per-billion, that may be applied toany nominal frequency, or in units of frequency, e.g. 100 Hz, asreferenced to a particular nominal frequency such as the nominal carrierfrequency. However expressed, those skilled in the art will alsoappreciate that frequency offsets measured by frequency-offsetestimation circuit 165 may be used to adjust the reference clock 159and/or PLL 158, as shown in FIG. 1, and/or to compensate one or moresignal processing functions in baseband signal processing circuit 160.

According to various embodiments of the present invention,frequency-offset estimation circuit 165 is configured to estimate thereceiver frequency offset relative to a received signal using theprocess generally illustrated in FIG. 2. As shown at block 210, aspatially and/or temporally stacked signal block may be formed bystacking two or more time-separated multi-branch samples of the receivedsignal, where the multiple signal branches may result from the use oftwo or more separate antennas, from multiple sampling phases of anover-sampled signal, or both. As will be explained in considerably moredetail below, the temporal and/or spatial correlation of the receivedsignal can be exploited to estimate the frequency offset. In particular,a maximum-likelihood joint estimate of the receiver frequency offset andthe spatial/temporal covariance of the temporally stacked signal block,de-rotated by the receiver frequency offset, can be computed, as shownat block 220. In some embodiments of the invention, the estimatedreceiver frequency offset may be used to adjust a frequency reference,e.g., by adjusting reference clock 159 or PLL circuit 158, or both, asshown at block 230. In some embodiments, the estimated receiverfrequency offset may be used to compensate baseband signal processing onthe received signal samples, such as by digitally de-rotating one ormore digital samples of the received signal. In some embodiments, acombination of these techniques may be used.

Following is a detailed derivation of exemplary formulas for use inestimating frequency offset in a received signal according to severalembodiments of the present invention. The derivation below is presentedfor explanatory purposes, and is not intended to be limiting, as thoseskilled in the art will appreciate that equivalent formulations and/orminor variations of the formulations presented herein may be used,according to the general techniques described above, to estimatereceiver frequency offset in various embodiments of the presentinvention.

First, consider the following complex-valued baseband model of amulti-branch signal received over a burst of N samples:

$\begin{matrix}{{{r_{c}\lbrack n\rbrack} = {\begin{bmatrix}{r_{c,1}\lbrack n\rbrack} \\{r_{c,2}\lbrack n\rbrack} \\M \\{r_{c,N_{r}}\lbrack n\rbrack}\end{bmatrix} = {^{{j\alpha}_{0}{({n - n_{0}})}}\left\{ {{u_{c}\lbrack n\rbrack} + {v_{c}\lbrack n\rbrack}} \right\}}}},} & (1)\end{matrix}$

for n=0, 1, L, N−1, where α₀ denotes the true (real-valued) relativefrequency offset with respect to the baud rate (in radian), n₀ denotes aconstant index pointing to the middle of a burst, r_(c,i)[n] denotes thei-th received signal branch, {u_(c)[n]} denotes the complex-valueddesired signal, and {v_(c)[n]} denotes a complex-valuedinterference-plus-noise vector process. Note that the multiple signalbranches may come from different physical antennas, different samplingphases of an oversampled signal, or both.

Now, let r[n, α] denote the received signal vector de-rotated by afrequency offset of α, i.e.:

r[n, α]≡e^(−jα{n−n) ⁰ ^(})r_(c)[n].   (2)

Also, let:

r_(M)[n, α]≡vec([r[n, α], r[n−1, α], L r[n−M, α]])   (3)

be the vector formed by stacking {r[k, α]}_(k−n−M) ^(n) in columns,where M denotes the “model order”, and, using MATLAB™ notation,vec(A)≡A(:) for any matrix A. Similarly, let:

r_(M)[n]≡vec([r_(c)[n], r_(c)[n−1], L r_(c)[n−M]])   (4)

be a vector formed by stacking {r_(c)[k]}_(k=n−M) ^(n) in columns; let

v_(M)[n]≡vec([v_(c)[n], v_(c)[n−1, ], L v_(c)[n−M]])   (5)

denotes the corresponding stacked noise vector, and let

u_(M)[n]≡vec([u_(c)[n], u_(c)[n−1,], L, u_(c)[n−M]])   (6)

denote the corresponding stacked desired-signal vector.

For purposes of deriving an estimate of the receiver frequency offset,we assume that the de-rotated received signal vector r r_(M)[n, α] has acomplex Gaussian distribution with a mean of zero. Those skilled in theart will recognize that this assumption holds for (at least) OrthogonalFrequency-Division Multiplexing (OFDM) signals, such as those used inLTE systems, as well as for signals modulated according to standards forGSM and EDGE. Further, we define a (spatial) covariance matrix:

Q≡E{r_(M)[n, α]r_(M)[n, α]^(H)}.   (7)

Given the stacked received signal vectors {r_(M)[n]}_(n=M) ^(N−1) forthe burst of N samples and the preceding assumptions and definitions,the maximum-likelihood joint estimate of the frequency offset α and thespatial covariance matrix Q can be computed. In other words, a frequencyoffset {circumflex over (α)}_(ML) and {circumflex over (Q)}_(ML) may becomputed to maximize the log-likelihood of the joint estimate, i.e., sothat:

$\begin{matrix}{\mspace{79mu} {{\left( {{\hat{\alpha}}_{ML},{\hat{Q}}_{ML}} \right) \equiv {\underset{({\alpha,Q})}{{argmax}\;}{{ll}\left( {\alpha,Q} \right)}}},\mspace{79mu} {{where}\text{:}}}} & (8) \\{{{{ll}\left( {\alpha,Q} \right)} \equiv {{{- \left( {N - M} \right)}\log \; \det \; Q} - {\sum\limits_{n = m}^{N - 1}\; {{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{H}Q^{- 1}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}}}} = {{{- \left( {N - M} \right)}\log \; \det \; Q} - {{tr}\left\{ {Q^{- 1}{\sum\limits_{n = M}^{N - 1}\; {{r_{M}\left\lbrack {n,\alpha} \right\rbrack}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{H}}}} \right\}}}} & (9)\end{matrix}$

is the log-likelihood function of α and Q, given the received signalvectors {r_(M)[n]}_(n=M) ^(N−1). (The function tr{A} represents thetrace of matrix A, while dct A is the determinant of A.)

From Equation (9), it can be shown that the best Q for any given α isgiven by

$\begin{matrix}{{{\hat{Q}(\alpha)} \equiv {\underset{Q}{argmin}\mspace{11mu} {{ll}\left( {\alpha,Q} \right)}}} = {\frac{1}{N - M}{\sum\limits_{n = M}^{N - 1}\; {{r_{M}\left\lbrack {n,\alpha} \right\rbrack}{{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{H}.}}}}} & (10)\end{matrix}$

For any given Q, if F(Q) denotes a square-root matrix of the inverse ofQ, such that Q⁻¹=F(Q)^(H)F(Q), then the best a for the given Q is givenby:

$\begin{matrix}{{{\hat{\alpha}(Q)} \equiv {\underset{\alpha}{argmin}{\sum\limits_{n = M}^{N - 1}\; {{{F(Q)}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}}^{2}}}},} & (11)\end{matrix}$

which must satisfy:

$\begin{matrix}{{\left. {\sum\limits_{n = M}^{N - 1}\; {{Re}\left\{ {\left\lbrack {{F(Q)}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}} \right\rbrack^{H}{F(Q)}\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha}} \right\}}} \right|_{\alpha = {\hat{\alpha}{(Q)}}} = 0},{{where}\text{:}}} & (12) \\{{\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha} = {{{blkdiag}\left( \left\{ {{j\left( {n - k - n_{0}} \right)}^{- {{j\alpha}{({n - k - n_{0}})}}}I_{N_{r}}} \right\}_{k = 0}^{M} \right)}{r_{M}\lbrack n\rbrack}}},} & (13)\end{matrix}$

in which blkdiag({A_(k)}_(k=0) ^(M)) denotes a block diagonal matrixwith matrices {A_(k)}_(k=0) ^(M) on the diagonal.

The solution of Equation (12) as a function of a does not have a closedform expression in general. However, since the frequency offset (inHertz) is typically much smaller than the baud rate, then the relativefrequency offset α is small, and the rotation e^(jθ) can be wellapproximated by the first few terms of its Taylor series expansion givenby

$\begin{matrix}{^{j\theta} = {1 + {j\theta} - \frac{\theta^{2}}{2} + \ldots}} & (14)\end{matrix}$

If only the first two terms of Equation (14) are used (i.e., a linearapproximation of e^(jθ)), then:

$\begin{matrix}{{{r_{M}\left\lbrack {n,\alpha} \right\rbrack} \cong {{r_{M}\lbrack n\rbrack} - {{{j\alpha}\left( {K_{n - M}^{n} \otimes I_{N_{r}}} \right)}{r_{M}\lbrack n\rbrack}}}},{and}} & (15) \\{{\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha} \cong {\left\lbrack {{j\; {K_{n - M}^{n} \otimes I_{N_{r}}}} + {{\alpha \left( K_{n - M}^{n} \right)}^{2} \otimes I_{N_{r}}}} \right\rbrack {r_{M}\lbrack n\rbrack}}},{where}} & (16) \\{K_{n - M}^{n} \equiv {\begin{pmatrix}{n - n_{0}} & 0 & K & 0 \\0 & {n - 1 - n_{0}} & O & M \\M & O & O & 0 \\0 & K & 0 & {n - M - n_{0}}\end{pmatrix}.}} & (17)\end{matrix}$

I_(N) denotes an N_(r)×N_(r) identity matrix, and {circle around (×)}denotes the Kronecker product.

If

x _(M) [n]=j(K _(n−M) ^(n)

I _(N) _(r) )r _(M) [n]  (18)

and

z _(M) [n]=((K _(n−M) ^(n))²

I _(N) _(r) )r _(M) [n],   (19)

then it follows that the necessary condition in Equation (12) reducesto:

$\begin{matrix}{{{{{\hat{\alpha}(Q)}^{2}{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q - 1}} + {{\hat{\alpha}(Q)}\left\lbrack {{{x_{M}\lbrack n\rbrack}}_{Q^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}} \right\rbrack} - {\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}} \cong 0}\mspace{79mu} {where}} & (20) \\{{{{\langle{{a\lbrack n\rbrack},{b\lbrack n\rbrack}}\rangle}_{Q^{- 1}} \equiv {\sum\limits_{n = {L + M}}^{N - 1}\; {{a\lbrack n\rbrack}^{H}{F(Q)}^{H}{F(Q)}{b\lbrack n\rbrack}}}} = {\sum\limits_{n = {L + M}}^{N - 1}\; {{a\lbrack n\rbrack}^{H}Q^{- 1}{b\lbrack n\rbrack}}}}\mspace{79mu} {and}\mspace{79mu} {{{a\lbrack n\rbrack}}_{Q^{- 1}}^{2} \equiv {{\langle{{a\lbrack n\rbrack},{a\lbrack n\rbrack}}\rangle}_{Q^{- 1}}.}}} & (21)\end{matrix}$

Solving Equation (20) yields:

$\begin{matrix}{{\hat{\alpha}(Q)} \cong {\frac{{{x_{M}\lbrack n\rbrack}}_{Q^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{2{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{\quad{\left\lbrack {\sqrt{1 + \frac{4{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{\left( {{{x_{M}\lbrack n\rbrack}}_{Q^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}} \right)^{2}}} - 1} \right\rbrack.}}}} & (22)\end{matrix}$

(Note that the other root of Equation (20) violates the assumption ofsmall α.)

Using the approximation √{square root over (1+x)}≈1+(½)x, Equation (22)simplifies to:

$\begin{matrix}{{\hat{\alpha}(Q)} \cong {\frac{{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}.}} & (23)\end{matrix}$

Those skilled in the art will appreciate that the formulation inEquation (10), which provides the best Q for any given α, may be usedalong with either Equation (22) or (23), which provide the best α for agiven Q, in an iterative manner to compute an approximatemaximum-likelihood estimate of the frequency offset. This approach isgenerally illustrated in the process flow diagram of FIG. 3, which maybe implemented, for example, using one or more processing circuits in awireless device, such as the baseband signal processing circuits 160pictured in the wireless transceiver 150 of FIG. 1.

The illustrated process begins with the initialization of afrequency-offset estimate {circumflex over (α)}⁽⁰⁾ and iteration indexi=0, as shown at block 310. In some embodiments, {circumflex over(α)}⁽⁰⁾ may be initialized at zero if no prior information about thefrequency offset is known. In other embodiments, a priorfrequency-offset estimate derived from a distinct estimation process,such as a coarse frequency-offset estimation process, may be used as aninitial estimate for the frequency offset. In some embodiments, theoutput from a previous blind estimation using the techniques describedherein may be used to initialize the frequency-offset estimate for asubsequent blind estimation process.

In any event, the process continues, as shown at block 320, with thegeneration of a spatially and temporally stacked signal block formed bystacking two or more (M+1, where M is a positive integer) multi-branchsignal samples from a received burst. Using the notation adopted above,the spatially stacked signal block may be given by:

r _(M) [n]=vec(r _(c) [n], r _(c) [n−1], L, r _(c) [n−M]).   (24)

A de-rotated received signal block is then computed, as shown at block330, from the stacked signal block and the current frequency-offsetestimate. Using the notation from the previous discussion, thede-rotated received signal block is given by:

r _(M) [n, {circumflex over (α)} ^((i))]=[diag(e^(−{circumflex over (α)}) ^((i)) ^((n−n) ⁰ ⁾ , e^(−{circumflex over (α)}) ^((i)) ^((n−1−n) ⁰ ⁾ , Λ, e^(−{circumflex over (α)}) ^((i)) ^((n−M−n) ⁰ ⁾)

I _(N) _(r) ]r _(M) [n].   (25)

An estimate of the signal covariance matrix Q is then calculateddirectly from the de-rotated received signal block, as shown at block340. In particular, given a series of de-rotated received signal blockscorresponding to samples M . . . N−1 of the received signal, thisestimate may be calculated according to:

$\begin{matrix}{{{\hat{Q}}^{(i)} = {\frac{1}{N - M}{\sum\limits_{n = M}^{N - 1}{{r_{M}\left\lbrack {n,{\hat{a}}^{(i)}} \right\rbrack}{r_{M}\left\lbrack {n,{\hat{a}}^{(i)}} \right\rbrack}^{H}}}}},} & (26)\end{matrix}$

and represents the best estimate of Q given the current estimate{circumflex over (α)}^((i)) for the frequency offset.

As shown at block 350, the current estimate of Q, i.e., {circumflex over(Q)}^((i)), is then used to update the frequency-offset estimate. Asshown above, in some embodiments this may be done by first calculating

x _(M) [n]=j(K _(n−M) ^(n)

I _(N) _(r) )r _(M) [n]  (27)

and

z _(M) [n]=((K _(n−M) ^(n))²

I _(N) _(r) )r _(M) [n],   (28)

and then using one of the approximations for {circumflex over (α)}^((i))given above, i.e.,

$\begin{matrix}{\mspace{79mu} {{\hat{\alpha}}^{({i + 1})} = \frac{{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{{(Q^{(i)})}^{- 1}}}{{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{(Q^{(i)})}^{- 1}}}}} & (29) \\{\mspace{79mu} {or}} & \; \\{{{{\hat{\alpha}}^{({i + 1})} = \frac{{{x_{M}\lbrack n\rbrack}}_{{({\hat{Q}}^{(i)})}^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}{2{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}}\quad}{\quad{\left\lbrack {\sqrt{1 + \frac{4{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}{\left( {{{x_{M}\lbrack n\rbrack}}_{{({\hat{Q}}^{(i)})}^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}} \right)^{2}}} - 1} \right\rbrack.}}} & (30)\end{matrix}$

Of course, those skilled in the art will appreciate that otherapproximations of the general formulation for {circumflex over(α)}^((i)) given in Equation (11) above may be used, in someembodiments.

The estimates for Q and α may be repeated, thus converging upon themaximum-likelihood values for each, until a pre-determined criteria issatisfied. In some embodiments, this criteria may simply be thecompletion of a pre-determined maximum number of iterations, as shown atblocks 360 and 370. In some embodiments, the iteration may be terminatedwhen an incremental change from one iteration to the next is smallerthan a pre-determined threshold, as shown at block 380. For instance, alog-likelihood value may be computed for each iteration according toEquation (9), if the change from one iteration to the next is smallerthan a pre-determined threshold, then the process is terminated.Otherwise, the iteration index i is incremented and the updating processis repeated.

As noted above, some receivers treat the in-phase and quadraturecomponents of a received signal as separate “spatial” dimensions, orbranches, and apply more general two-dimensional “spatial” operations onthe received signal. The techniques described above may be modifiedslightly to accommodate the separate handling of the in-phase andquadrature components. As will be appreciated by those skilled in theart, this approach exploits the spatial correlation across in-phase andquadrature domains in addition to the temporal correlations betweensuccessive samples. Thus, substantial performance gains can be realizedwhen the received signal's modulation constellation is one-dimensionalon the in-phase/quadrature plane.

Accordingly, FIG. 4 illustrates a simplified block diagram of thereceiver portion of a wireless transceiver 450. In contrast to thewireless transceiver 150 illustrated in FIG. 1, which had two receiveantennas, wireless transceiver 450 includes only a single receiveantenna. Of course, those skilled in the art will appreciate that one ormore additional receive antennas may be used in other embodiments. Inany event, the receive antenna of wireless transceiver 450 provides thereceived signal to duplexing circuit 452, which in turn provides thereceived signal to receiver front-end circuit 456. As discussed abovewith respect to FIG. 1, those skilled in the art will appreciate thatduplexer circuit 452 may comprise one or more filters and/or switches,depending on the wireless standard, and serves to alternately connectthe antenna to the receiver (RX) front-end circuit 456 and a transmitter(TX) analog section, or to isolate signals generated by the transmittersection from the RX front-end circuit 456, or both. As pictured, RXfront-end circuit 456 includes a low-noise amplifier and in-phase andquadrature mixers, driven by in-phase and quadrature local oscillatorsignals from phase-locked loop circuit 458. As was discussed above withrespect to FIG. 1, those skilled in the art will appreciate that thetransmitter section and RX front-end circuit 456 of transceiver 450further includes additional components that are not illustrated in FIG.4, such as one or more power amplifiers, low-noise amplifiers, filters,mixers, modulators, analog-to-digital converters (ADCs),digital-to-analog converters (DACs), and the like, configured accordingto well-known techniques appropriate for the wireless standard(s) andfrequency band(s) supported by wireless transceiver 450. Accordingly,further details of these circuits, which details are not necessary foran understanding of the present invention, are not provided herein.

RX front-end 456 is supplied with in-phase (0°) and quadrature (90°)reference signals from phase-locked loop (PLL) circuit 458, which inturn is driven by a reference clock oscillator 459. Again, designdetails for various suitable phase-locked loop circuits and referenceoscillator circuits are well known to those skilled in the art, and arenot provided here. Baseband signal processing circuit 460 thus receivesseparate in-phase (I) and quadrature (Q) components of the receivedsignal from RX front-end circuit 456; these I and Q components may bedigitized using matching analog-to-digital converters (not shown)located in RX front-end circuit 456 or baseband signal processingcircuit 460. Among its other functions, baseband signal processingcircuit 460 performs demodulation and decoding processes according toconventional techniques, which may include single-antenna interferencecancellation (SAIC) techniques that treat the I and Q components of thereceived signal as though they come from two different antenna elements.

In the processes described in detail above, the temporal (block)correlation of the received signal was exploited to derive an estimateof the receiver frequency offset. As will be shown in detail below,other embodiments of the present invention, including, but not limitedto, single-antenna receiver structures such as that pictured in FIG. 4,may exploit the spatial correlation between the in-phase and quadraturecomponents of the received signal, as well as temporal correlations, toestimate the frequency offset. As those skilled in the art willappreciate, the below derivation of formulations for thetemporal/spatial covariance and the estimated frequency offset is verysimilar to that described in detail above. The main difference is thatthe real and imaginary parts of a complex-signal are treated as atwo-dimensional, real-valued vector signal.

By stacking the real and imaginary part of each signal branch r_(c,i)[n]into a 2-by-1 vector, the signal model of Equation (1) can be expressedas:

r[n]=(I _(N) _(r)

Φ_(α) ₀ _((n−n) ₀ ₎){u[n]+v[n]},   (31)

for n=0, 1, L, N−1, where I_(N) _(r) denotes N_(r)×N_(r) identitymatrix, {circle around (×)} denotes the Kronecker product, and where

$\begin{matrix}{\Phi_{\theta} = {\begin{bmatrix}{\cos \; \theta} & {{- \sin}\; \theta} \\{\sin \; \theta} & {\cos \; \theta}\end{bmatrix}.}} & (32)\end{matrix}$

Accordingly, r[n] denotes a real-valued, multi-branch, received signalvector with dimension 2N_(r), u[n] denotes the corresponding real-valueddesired signal, and v[n] denotes a corresponding real-valuedinterference-plus-noise process.

Now, let r[n, α] denote the received signal vector de-rotated by afrequency offset α, i.e.:

r[n, α]≡(I_(N) _(r)

Φ_(α(n−n) ₀ ₎ ^(T))r[n].   (33)

Also, let

r_(M)[n, α]≡vec([r[n, α], r[n−1, α], L r[n−M, α]])   (34)

be the vector formed by stacking {r[k, α]}_(k=n−M) ^(n) in columns,where M denotes the “model order”, and, using MATLAB notation,vec(A)≡A(:) for any matrix A. The model order M may be any non-negativeinteger, i.e., 0 or above. Similarly, let

r_(M)[n]≡vec([r[n], r[n−1], L r[n−M]])   (35)

be a vector formed by stacking {r[k]}_(k=n−M) ^(n) in columns; let

v_(M)[n]≡vec([v[n], v[n−1], L v[n−M]])   (36)

denote the corresponding stacked noise vector; and let

u_(M)[n]≡vec([u[n], u[n−1], L, u[n−M]])   (37)

denote the corresponding stacked training vector.

In the special case where M=0, Equations 34-37 can be simplified to:

r_(M)[n, α]≡r[n, α]  (38)

r _(M)[n]≡r[n]  (39)

v _(M)[n]≡v[n]  (40)

u _(M)[n]≡u[n]  (41)

As in the earlier case, we assume that the de-rotated received signalvector r_(M)[n, α] has a real-valued Gaussian distribution with a meanof zero and a (spatial) covariance matrix Q≡E{ r_(M)[n, α]r_(M)[n,α]^(H)}. Given these assumptions, then the maximum-likelihood jointestimate of the frequency offset α and the spatial covariance matrix Qcan be computed. In other words, an iterative process can be used tofind:

$\begin{matrix}{{\left( {{\hat{\alpha}}_{ML},{\hat{Q}}_{ML}} \right) \equiv {\underset{({\alpha,Q})}{argmax}\mspace{11mu} {{ll}\left( {\alpha,Q} \right)}}},} & (42) \\{where} & \; \\\begin{matrix}{{{ll}\left( {\alpha,Q} \right)} \equiv {{{- \left( {N - M} \right)}\log \; {d{et}}\; Q} - {\sum\limits_{n = {L + M}}^{N - 1}{{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{T}Q^{- 1}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}}}} \\{= {{{- \left( {N - M} \right)}{{logd}{et}}\; Q} -}} \\{{{tr}\left\{ {Q^{- 1}{\sum\limits_{n = M}^{N - 1}{{r_{M}\left\lbrack {n,\alpha} \right\rbrack}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{T}}}} \right\}}}\end{matrix} & (43)\end{matrix}$

is the log-likelihood function of α and Q, given the stacked receivedsignal vectors {r_(M)[n]}_(n=M) ^(N−1) for the burst of N samples.

From Equation (43), it can be shown that the best Q for any given α isgiven by:

$\begin{matrix}{{{\hat{Q}(\alpha)} \equiv {\underset{Q}{\arg \min}\mspace{11mu} {{ll}\left( {\alpha,Q} \right)}}} = {\frac{1}{N - M}{\sum\limits_{n = M}^{N - 1}{{r_{M}\left( {n,\alpha} \right)}{{r_{M}\left\lbrack {n,\alpha} \right\rbrack}^{T}.}}}}} & (44)\end{matrix}$

For any given Q, if F(Q) denote a square-root matrix of the inverse of Qsuch that Q⁻¹=F(Q)^(H)F(Q), then the best α for the given Q reduces to:

$\begin{matrix}{{{\hat{\alpha}(Q)} \equiv {\underset{\alpha}{\arg \min}{\sum\limits_{n = {L + M}}^{N - 1}{{{F(Q)}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}}^{2}}}},} & (45)\end{matrix}$

which must satisfy:

$\begin{matrix}{{{\mspace{79mu} {\sum\limits_{n = M}^{{N - 1}\;}{{Re}\left\{ {\left\lbrack {{F(Q)}{r_{M}\left\lbrack {n,\alpha} \right\rbrack}} \right\rbrack^{H}{F(Q)}\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha}} \right\}}}}_{\alpha = {\hat{\alpha}{(Q)}}} = 0},} & (46) \\{\mspace{79mu} {where}} & \; \\{\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha} = {{{blkdiag}\left( \left\{ {{I_{N_{r}} \otimes \left( {n - k - n_{0}} \right)}\Phi_{\alpha {({n - k - n_{0}})}}^{T}J} \right\}_{k = 0}^{M} \right)}{r_{M}\lbrack n\rbrack}}} & (47) \\{\mspace{79mu} {{{where}\mspace{14mu} J} \equiv \begin{pmatrix}0 & {- 1} \\1 & 0\end{pmatrix}}} & \;\end{matrix}$

and blkdiag({A_(k)}_(k=0) ^(M)) denotes a block diagonal matrix withmatrices {A_(k)}_(k=0) ^(M) on its diagonal.

The solution of Equation (47) as a function of a does not have closedform expression in general. However, since the frequency offset in Hertzis typically much smaller than the baud rate, then the relativefrequency offset α is small and the rotational matrix Φ_(θ) can be wellapproximated by the first few terms of its Taylor series expansion givenby

$\begin{matrix}{\Phi_{\theta} = {I + {\theta \; J} - {\frac{\theta^{2}}{2}I} + \ldots}} & (48)\end{matrix}$

Using only the first two terms of Equation (43) (i.e., a linearapproximation of Φ_(θ)), then:

$\begin{matrix}{{{r_{M}\left\lbrack {n,\alpha} \right\rbrack} \cong {{r_{M}\lbrack n\rbrack} - {{\alpha \left( {K_{n - M}^{n} \otimes I_{N_{r}} \otimes J} \right)}{r_{M}\lbrack n\rbrack}}}},} & (49) \\{and} & \; \\{{\frac{\partial{r_{M}\left\lbrack {n,\alpha} \right\rbrack}}{\partial\alpha} \cong {\left\lbrack {{K_{n - M}^{n} \otimes I_{N_{r}} \otimes J} + {{\alpha \left( K_{n - M}^{n} \right)}^{2} \otimes I_{2N_{r}}}} \right\rbrack {r_{M}\lbrack n\rbrack}}},} & (50) \\{where} & \; \\{K_{n - M}^{n} \equiv {\begin{pmatrix}{n - n_{0}} & 0 & K & 0 \\0 & {n - 1 - n_{0}} & O & M \\M & O & O & 0 \\0 & K & 0 & {n - M - n_{0}}\end{pmatrix}.}} & (51)\end{matrix}$

If x_(M)[n]=(K_(n−M) ^(n)

I_(N) _(r)

J)r_(M)[n] and z_(M)[n]=((K_(n−M) ^(n))²

I_(2N) _(r) )r_(M)[n], then it follows that the necessary condition inEquation (42) reduces to:

{circumflex over (α)}(Q)²

x_(M)[n], z_(M)[n]

_(Q) ⁻¹ +{circumflex over (α)}(Q)[∥x_(M)[n]∥_(Q) ⁻¹ ²−

x_(M)[n], z_(M)[n]

_(Q) ⁻¹ ]−

r_(M)[n], x_(M)[n]

_(Q) ⁻¹ ≅0,   (52)

where

$\begin{matrix}{\begin{matrix}{{\langle{{a\lbrack n\rbrack},{b\lbrack n\rbrack}}\rangle}_{Q^{- 1}} \equiv {\sum\limits_{n = {L + M}}^{N - 1}{{a\lbrack n\rbrack}^{H}{F(Q)}^{H}{F(Q)}{b\lbrack n\rbrack}}}} \\{= {\sum\limits_{n = {L + M}}^{N - 1}{{a\lbrack n\rbrack}^{H}Q^{- 1}{b\lbrack n\rbrack}}}}\end{matrix}{and}{{{a\lbrack n\rbrack}}_{Q^{- 1}}^{2} \equiv {{\langle{{a\lbrack n\rbrack},{a\lbrack n\rbrack}}\rangle}_{Q^{- 1}}.}}} & (53)\end{matrix}$

Solving Equation (52) yields the best estimate of the frequency offsetfor a given covariance Q:

$\begin{matrix}{{\hat{\alpha}(Q)} \cong {\frac{{{x_{M}\lbrack n\rbrack}}_{Q^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{2{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{\quad{\left\lbrack {\sqrt{1 + \frac{4{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{\left( {{{x_{M}\lbrack n\rbrack}}_{Q^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}} \right)^{2}}} - 1} \right\rbrack.}}}} & (54)\end{matrix}$

(Note that the other root of Equation (52) violates the assumption ofsmall α.)

Using the approximation √{square root over (1+x)}≈1+(½)x, Equation (54)simplifies to

$\begin{matrix}{{\hat{\alpha}(Q)} \cong {\frac{{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}{{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{Q^{- 1}}}.}} & (55)\end{matrix}$

Those skilled in the art will readily appreciate that the formulation inEquation (44) which provides the best Q for any given α, may be usedalong with either Equation (54) or (55), which provide the best αfor agiven Q, in an iterative manner, to compute an approximatemaximum-likelihood estimate of the frequency offset. This approach isgenerally the same as discussed above in connection with FIG. 3, exceptthat it is this time based on the alternative formulations discussedimmediately above.

Accordingly, a spatially and/or temporally stacked signal block isformed according to:

r _(M) [n]=vec(r[n], r[n−1], L, r[n−M]),   (56)

and a de-rotated received signal block computed according to:

r _(M[n, {circumflex over (α)}) ^((i))]=blkdiag(I _(N) _(r)

Φ_({circumflex over (α)}) _((i)) _((n−n) ₀ ₎ ^(T) , I _(N) _(r)

Φ_({circumflex over (α)}) _((i)) _((n−1−n) ₀ ₎ ^(T) , Λ, I _(N) _(r)

Φ_({circumflex over (α)}) _((i)) _(n−M−n) ₀ ₎ ^(T))r _(M) [m].   (57)

Given an initial estimate of the receiver frequency offset, then anestimate of the signal covariance matrix is computed:

$\begin{matrix}{{{\hat{Q}}^{(i)} = {\frac{1}{N - L - M}{\sum\limits_{n = {L + M}}^{N - 1}{{r_{M}\left\lbrack {n,{\hat{\alpha}}^{(i)}} \right\rbrack}{r_{M}\left\lbrack {n,{\hat{\alpha}}^{(i)}} \right\rbrack}^{T}}}}},} & (58)\end{matrix}$

and the receiver frequency-offset estimate updated according to:

$\begin{matrix}{\mspace{79mu} {{\hat{\alpha}}^{i + 1} = \frac{{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{{(Q^{(i)})}^{- 1}}}{{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{(Q^{(i)})}^{- 1}}}}} & (59) \\{\mspace{79mu} {or}} & \; \\{{\hat{\alpha}}^{({i + 1})} \cong {\frac{{{x_{M}\lbrack n\rbrack}}_{{({\hat{Q}}^{(i)})}^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}{2{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}{\quad{\left\lbrack {\sqrt{1 + \frac{4{\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}{\langle{{r_{M}\lbrack n\rbrack},{x_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}}{\left( {{{x_{M}\lbrack n\rbrack}}_{{({\hat{Q}}^{(i)})}^{- 1}}^{2} - {\langle{{x_{M}\lbrack n\rbrack},{z_{M}\lbrack n\rbrack}}\rangle}_{{({\hat{Q}}^{(i)})}^{- 1}}} \right)^{2}}} - 1} \right\rbrack,}}}} & (60) \\{\mspace{79mu} {where}} & \; \\{\mspace{79mu} {{x_{M}\lbrack n\rbrack} = {\left( {K_{n - M}^{n} \otimes I_{N_{r}} \otimes J} \right){r_{M}\lbrack n\rbrack}}}} & (61) \\{\mspace{79mu} {and}} & \; \\{\mspace{79mu} {{z_{M}\lbrack n\rbrack} = {\left( {\left( K_{n - M}^{n} \right)^{2} \otimes I_{2N_{r}}} \right){{r_{M}\lbrack n\rbrack}.}}}} & (62)\end{matrix}$

As was described above, this estimation process may be repeated until apre-determined stopping criterion is satisfied, such as the completionof a maximum number of iterations, or convergence of the estimates to apoint at which the incremental change in the log-likelihood of theestimates is smaller than a pre-determined threshold).

Those skilled in the art will appreciate that an advantage of thetechniques described above is to that a receiver can compute an estimateof the frequency offset of the received signal without using anytraining signal or running computationally intensive equalization ordemodulation algorithms. This allows a mobile terminal to periodicallycompute frequency-offset estimates even when there is neither trainingsignal nor data-bearing signal transmitted to it. Of course, in someembodiments these techniques may be used in a receiver along with priortechniques that depend on training signals and/or demodulated signals.In some embodiments, the techniques described above may be selectivelyemployed depending on whether training signals and/or demodulatedsignals are available.

With these and other variations and extensions in mind, those skilled inthe art will appreciate that the preceding descriptions of variousembodiments of methods and apparatus estimating receiver frequencyoffset in a communications receiver are given for purposes ofillustration and example. As suggested above, one or more of thetechniques discussed above, including the process flows illustrated inFIGS. 2 and 3, may be carried out in a wireless receiver comprising oneor more appropriately configured processing circuits, which may in someembodiments be embodied in one or more application-specific integratedcircuits (ASICs). In some embodiments, these processing circuits maycomprise one or more microprocessors, microcontrollers, and/or digitalsignal processors programmed with appropriate software and/or firmwareto carry out one or more of the processes described above, or variantsthereof. In some embodiments, these processing circuits may comprisecustomized hardware to carry out one or more of the functions describedabove. Other embodiments of the invention may include computer-readabledevices, such as a programmable flash memory, an optical or magneticdata storage device, or the like, encoded with computer programinstructions which, when executed by an appropriate processing device,cause the processing device to carry out one or more of the techniquesdescribed herein for estimating receiver frequency offset in acommunications receiver. Those skilled in the art will recognize, ofcourse, that the present invention may be carried out in other ways thanthose specifically set forth herein without departing from essentialcharacteristics of the invention. The present embodiments are thus to beconsidered in all respects as illustrative and not restrictive, and allchanges coming within the meaning and equivalency range of the appendedclaims are intended to be embraced therein.

1. A method of estimating receiver frequency offset in a communicationsreceiver, the method comprising: forming a spatially stacked signalblock comprising real-valued in-phase and quadrature samples of acomplex-valued signal received on one or more branches; and computing amaximum-likelihood joint estimate of the receiver frequency offset andthe spatial covariance of the spatially stacked signal block de-rotatedby the receiver frequency offset.
 2. The method of claim 1, furthercomprising stacking two or more spatially stacked signal blocksseparated in time to form a temporally and spatially stacked signalblock.
 3. The method of claim 1, wherein computing a maximum-likelihoodjoint estimate of the receiver frequency offset and the spatialcovariance of the spatially stacked signal block de-rotated by thereceiver frequency offset comprises, for each of two or more iterations:computing a de-rotated received signal block from the spatially stackedsignal block, using a current estimate of the receiver frequency offset;estimating a signal covariance matrix from the de-rotated receivedsignal block; and updating the current estimate of the receiverfrequency offset as a function of the estimated signal covariancematrix.
 4. The method of claim 3, wherein the two or more iterationscomprise an initial iteration for which the current estimate of thereceiver frequency offset is initialized to one of: zero; a coarsefrequency offset obtained from a coarse frequency-offset estimationprocess; and a prior maximum-likelihood estimate of receiver frequencyoffset.
 5. The method of claim 3, further comprising iterativelyupdating the current estimate of the receiver frequency offset for apre-determined maximum number of iterations.
 6. The method of claim 3,further comprising computing a log-likelihood of the joint estimate ofthe receiver frequency offset and the spatial covariance, for eachiteration, and iteratively updating the current estimate of the receiverfrequency offset until the incremental change in the computedlog-likelihoods between successive iterations falls below apre-determined threshold.
 7. A wireless communications receivercomprising one or more processing circuits configured to: form aspatially stacked signal block comprising real-valued in-phase andquadrature samples of a complex-valued signal received on one or morebranches; and compute a maximum-likelihood joint estimate of thereceiver frequency offset and the spatial covariance of the spatiallystacked signal block de-rotated by the receiver frequency offset.
 8. Thewireless communications receiver of claim 7, wherein the processingcircuits are further configured to stack two or more spatially stackedsignal blocks separated in time to form a temporally and spatiallystacked signal block.
 9. The wireless communications receiver of claim8, wherein the one or more processing circuits are configured to computethe maximum-likelihood joint estimate of the receiver frequency offsetand the spatial covariance of the spatially stacked signal blockde-rotated by the receiver frequency offset by, for each of two or moreiterations: computing a de-rotated received signal block from thespatially stacked signal block, using a current estimate of the receiverfrequency offset; estimating a signal covariance matrix from thede-rotated received signal block; and updating the current estimate ofthe receiver frequency offset as a function of the estimated signalcovariance matrix.
 10. The wireless communications receiver of claim 9,wherein the two or more iterations comprise an initial iteration andwherein the one or more processing circuits are configured to initializethe current estimate of the receiver frequency offset for the initialiteration to one of: zero; a coarse frequency offset obtained from acoarse frequency-offset estimation process; and a priormaximum-likelihood estimate of receiver frequency offset.
 11. Thewireless communications receiver of claim 9, wherein the one or moreprocessing circuits are further configured to iteratively update thecurrent estimate of the receiver frequency offset for a pre-determinedmaximum number of iterations.
 12. The wireless communications receiverof claim 9, wherein the one or more processing circuits are furtherconfigured to compute a log-likelihood of the joint estimate of thereceiver frequency offset and the spatial covariance, for eachiteration, and to iteratively update the current estimate of thereceiver frequency offset until the incremental change in the computedlog-likelihoods between successive iterations falls below apre-determined threshold.